module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 192,
"column": 8
} | {
"line": 192,
"column": 98
} | {
"line": 193,
"column": 8
} | [
{
"pp": "case inr\nV✝ : Type u_1\nG✝ : SimpleGraph V✝\nV : Type u_1\nG : SimpleGraph V\nn : ℕ\nhn : 2 < n\na : V\np : G.Walk a a\nhp₁ : p.IsCycle\nhp₂ : p.length = n\nx : ℕ\nhx : x < n\ny : ℕ\nhy : y < n\nhab✝ : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩\nhne : x ≠ y\nhle : x > y\nhab : n + ↑⟨y, hy⟩ - ↑⟨x, hx⟩ = 1\n⊢ G... | [
"case inr\nV✝ : Type u_1\nG✝ : SimpleGraph V✝\nV : Type u_1\nG : SimpleGraph V\nn : ℕ\nhn : 2 < n\na : V\np : G.Walk a a\nhp₁ : p.IsCycle\nhp₂ : p.length = n\nx : ℕ\nhx : x < n\ny : ℕ\nhy : y < n\nhab✝ : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩\nhne : x ≠ y\nhle : x > y\nhab : n + ↑⟨y, hy⟩ - ↑⟨x, hx⟩ = 1\n⊢ G.Adj p.suppo... | simp_rw [show x = n - 1 by lia, show y = 0 by lia, Fin.succ_mk, show n - 1 + 1 = n by lia] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 591,
"column": 57
} | {
"line": 591,
"column": 68
} | {
"line": 591,
"column": 69
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | [
"V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right\nleft right... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 593,
"column": 58
} | {
"line": 593,
"column": 69
} | {
"line": 593,
"column": 70
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | [
"V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right\nleft right... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 597,
"column": 8
} | {
"line": 597,
"column": 19
} | {
"line": 597,
"column": 20
} | [
{
"pp": "case refine_1.inr.inl.refine_3\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCove... | [
"case refine_1.inr.inl.refine_3\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsComplete... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 598,
"column": 6
} | {
"line": 598,
"column": 17
} | {
"line": 598,
"column": 18
} | [
{
"pp": "case refine_1.inr.inr\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompl... | [
"case refine_1.inr.inr\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 27
} | {
"line": 130,
"column": 0
} | [
{
"pp": "M : Type u_1\ns : Set M\ninst✝ : Group M\nu v : M\n⊢ (mulCayley s).Adj u v ↔ u ≠ v ∧ (u⁻¹ * v ∈ s ∨ v⁻¹ * u ∈ s)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"and_true",
"Monoid.toMulOneClass",
"congrAr... | [] | simp [mulCayley_adj', ← eq_inv_mul_iff_mul_eq (b := u), ← inv_mul_eq_iff_eq_mul (a := v),
and_or_left, exists_or] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 27
} | {
"line": 130,
"column": 0
} | [
{
"pp": "M : Type u_1\ns : Set M\ninst✝ : Group M\nu v : M\n⊢ (mulCayley s).Adj u v ↔ u ≠ v ∧ (u⁻¹ * v ∈ s ∨ v⁻¹ * u ∈ s)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"and_true",
"Monoid.toMulOneClass",
"congrAr... | [] | simp [mulCayley_adj', ← eq_inv_mul_iff_mul_eq (b := u), ← inv_mul_eq_iff_eq_mul (a := v),
and_or_left, exists_or] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 27
} | {
"line": 130,
"column": 0
} | [
{
"pp": "M : Type u_1\ns : Set M\ninst✝ : Group M\nu v : M\n⊢ (mulCayley s).Adj u v ↔ u ≠ v ∧ (u⁻¹ * v ∈ s ∨ v⁻¹ * u ∈ s)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"and_true",
"Monoid.toMulOneClass",
"congrAr... | [] | simp [mulCayley_adj', ← eq_inv_mul_iff_mul_eq (b := u), ← inv_mul_eq_iff_eq_mul (a := v),
and_or_left, exists_or] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 605,
"column": 4
} | {
"line": 605,
"column": 28
} | {
"line": 605,
"column": 29
} | [
{
"pp": "case refine_2\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ : ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.IsCompleteBetween ↑left ↑right\nleft ri... | [
"case refine_2\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ : ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.IsCompleteBetween ↑left ↑right\nleft right : Finset... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 15
} | {
"line": 116,
"column": 16
} | [
{
"pp": "case cons\nV : Type u_1\nHs : Set (SimpleGraph V)\nhHs : Hs.Nonempty\nh_dir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) Hs\nu✝ v✝ u v w : V\np : (sSup Hs).Walk v w\nH₁ : SimpleGraph V\nhH₁ : H₁ ∈ Hs\nih : ∀ e ∈ p.edges, e ∈ H₁.edgeSet\nH₂ : SimpleGraph V\nhH₂ : H₂ ∈ Hs\nh_adj : H₂.Adj u v\nH : SimpleGraph V\nhH... | [
"case cons\nV : Type u_1\nHs : Set (SimpleGraph V)\nhHs : Hs.Nonempty\nh_dir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) Hs\nu✝ v✝ u v w : V\np : (sSup Hs).Walk v w\nH₁ : SimpleGraph V\nhH₁ : H₁ ∈ Hs\nih : ∀ e ∈ p.edges, e ∈ H₁.edgeSet\nH₂ : SimpleGraph V\nhH₂ : H₂ ∈ Hs\nh_adj : H₂.Adj u v\nH : SimpleGraph V\nhH : H ∈ Hs\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 218,
"column": 86
} | {
"line": 221,
"column": 7
} | {
"line": 223,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nh : G.IsAcyclic\nu v w : V\np : G.Walk u v\nhp : p.IsPath\nhadj : G.Adj v w\nhsupp : w ∈ p.support\n⊢ w = p.penultimate",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"List.mem_reverse._simp_1",
"Eq.mpr",
"congrArg",
"Simpl... | [] | by
rw [← snd_reverse]
apply h.eq_snd_of_adj_start hp.reverse hadj
simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 73
} | {
"line": 291,
"column": 6
} | [
{
"pp": "case cons.inr\nV : Type u_1\nG : SimpleGraph V\nhG : G.IsAcyclic\nv w u' v' w✝ : V\nhead : G.Adj u' v'\ntail : G.Walk v' w✝\nih : List.IsChain (fun x1 x2 ↦ x1 ≠ x2) tail.edges → tail.IsPath\nh : List.IsChain (fun x1 x2 ↦ x1 ≠ x2) (cons head tail).edges\nhcc : (∀ y ∈ tail.edges.head?, s(u', v') ≠ y) ∧ L... | [
"case cons.inr\nV : Type u_1\nG : SimpleGraph V\nhG : G.IsAcyclic\nv w u' v' w✝ : V\nhead : G.Adj u' v'\ntail : G.Walk v' w✝\nih : List.IsChain (fun x1 x2 ↦ x1 ≠ x2) tail.edges → tail.IsPath\nh : List.IsChain (fun x1 x2 ↦ x1 ≠ x2) (cons head tail).edges\nhcc : (∀ y ∈ tail.edges.head?, s(u', v') ≠ y) ∧ List.IsChain ... | have := IsPath.mk' this |>.eq_snd_of_mem_edges (Sym2.eq_swap ▸ hhh) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 53,
"column": 14
} | {
"line": 53,
"column": 25
} | {
"line": 53,
"column": 26
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝ : ∃ G, p G\nG : SimpleGraph V\nhp : p G\n⊢ G ∈ {G | p G}",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"instFintypeSimpleGraphOfDecidableEq",
"Eq.mpr",
"Finset.mem_filter._simp_1",
"Finset.univ... | [
"V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝ : ∃ G, p G\nG : SimpleGraph V\nhp : p G\n⊢ p G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 55,
"column": 12
} | {
"line": 55,
"column": 23
} | {
"line": 55,
"column": 24
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝ : ∃ G, p G\nG : SimpleGraph V\nhp : p G\nG' : SimpleGraph V\nhp' : G' ∈ {G | p G}\nh : ∀ x' ∈ {G | p G}, #x'.edgeFinset ≤ #G'.edgeFinset\n⊢ p G'",
"ppTerm": "?m.133",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝ : ∃ G, p G\nG : SimpleGraph V\nhp : p G\nG' : SimpleGraph V\nhp' : G' ∈ {G | p G}\nh : ∀ x' ∈ {G | p G}, #x'.edgeFinset ≤ #G'.edgeFinset\n⊢ p G'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 55,
"column": 62
} | {
"line": 55,
"column": 73
} | {
"line": 55,
"column": 74
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝² : ∃ G, p G\nG : SimpleGraph V\nhp✝ : p G\nG' : SimpleGraph V\nhp' : G' ∈ {G | p G}\nh : ∀ x' ∈ {G | p G}, #x'.edgeFinset ≤ #G'.edgeFinset\nx✝¹ : SimpleGraph V\nx✝ : DecidableRel x✝¹.Adj\nhp : p x✝¹\n⊢ ?m.149 ∈ {G | p G}",
"ppTerm": "?m.... | [
"V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝² : ∃ G, p G\nG : SimpleGraph V\nhp✝ : p G\nG' : SimpleGraph V\nhp' : G' ∈ {G | p G}\nh : ∀ x' ∈ {G | p G}, #x'.edgeFinset ≤ #G'.edgeFinset\nx✝¹ : SimpleGraph V\nx✝ : DecidableRel x✝¹.Adj\nhp : p x✝¹\n⊢ p ?m.149"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 310,
"column": 41
} | {
"line": 310,
"column": 52
} | {
"line": 310,
"column": 53
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (fun p ↦ p.Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 15
} | {
"line": 96,
"column": 16
} | [
{
"pp": "n : ℕ\nV : Type u_1\nW : Type u_2\nH : SimpleGraph W\ninst✝ : Fintype V\nhc : Fintype.card V = n\ne : Fin n ≃ V\nG : SimpleGraph (Fin n)\nh : G ∈ {G | H.Free G}\n⊢ SimpleGraph.map (⇑e) G ∈ univ ∧ H.Free G",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"instFintypeSimpleGrap... | [
"n : ℕ\nV : Type u_1\nW : Type u_2\nH : SimpleGraph W\ninst✝ : Fintype V\nhc : Fintype.card V = n\ne : Fin n ≃ V\nG : SimpleGraph (Fin n)\nh : G ∈ {G | H.Free G}\n⊢ IsEmpty (H.Copy G)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 15
} | {
"line": 96,
"column": 16
} | [
{
"pp": "n : ℕ\nV : Type u_1\nW : Type u_2\nH : SimpleGraph W\ninst✝ : Fintype V\nhc : Fintype.card V = n\ne : V ≃ Fin n := Fintype.equivFinOfCardEq hc\nG : SimpleGraph V\nh : G ∈ {G | H.Free G}\n⊢ SimpleGraph.map (⇑e) G ∈ univ ∧ H.Free G",
"ppTerm": "?m.339",
"assigned": true,
"usedConstants": [
... | [
"n : ℕ\nV : Type u_1\nW : Type u_2\nH : SimpleGraph W\ninst✝ : Fintype V\nhc : Fintype.card V = n\ne : V ≃ Fin n := Fintype.equivFinOfCardEq hc\nG : SimpleGraph V\nh : G ∈ {G | H.Free G}\n⊢ IsEmpty (H.Copy G)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 106,
"column": 64
} | {
"line": 106,
"column": 75
} | {
"line": 106,
"column": 76
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nh : H.Free G\n⊢ G ∈ {G | H.Free G}",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"SimpleGraph.Free",
"Eq.mpr",
"Finset.mem_filter._simp_1",
"... | [
"V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nh : H.Free G\n⊢ IsEmpty (H.Copy G)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 319,
"column": 54
} | {
"line": 319,
"column": 65
} | {
"line": 319,
"column": 66
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (fun p ↦ p.Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 323,
"column": 49
} | {
"line": 323,
"column": 60
} | {
"line": 323,
"column": 61
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (fun p ↦ p.Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 331,
"column": 4
} | {
"line": 332,
"column": 45
} | {
"line": 333,
"column": 4
} | [
{
"pp": "case inr\nV : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝¹ : Nonempty V\ninhabited_h : Inhabited V\nthis✝ : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (fun p ↦ p.Is... | · rw [Sym2.eq_swap]
exact this y x h.symm (le_of_not_ge h') | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 336,
"column": 56
} | {
"line": 336,
"column": 67
} | {
"line": 336,
"column": 68
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (fun p ↦ p.Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 100,
"column": 4
} | {
"line": 101,
"column": 42
} | {
"line": 101,
"column": 43
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nK : Finset V\nleft✝ : K ⊆ univ\nhn : #K ≤ Fintype.card V\nhG : G.IsTuranMaximal #K\nh : G.CliqueFree #K\n⊢ ∃ a ∈ K, ∃ b ∈ K, a ≠ b ∧ ¬G.Adj a b",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"F... | [
"V : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nK : Finset V\nleft✝ : K ⊆ univ\nhn : #K ≤ Fintype.card V\nhG : G.IsTuranMaximal #K\nh : G.CliqueFree #K\n⊢ ∃ a ∈ K, ∃ b ∈ K, ¬a = b ∧ ¬G.Adj a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 60
} | {
"line": 107,
"column": 61
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\nr : ℕ\nhr : 0 < r\n⊢ ∃ H x, H.IsTuranMaximal r",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Adj",
"DecidableRel",
"Exists",
"id",
"instOfNatNat",
"_private.Mathlib.Combinatorics.Si... | [
"V : Type u_1\ninst✝ : Fintype V\nr : ℕ\nhr : 0 < r\n⊢ ∃ x, x.CliqueFree (r + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 388,
"column": 67
} | {
"line": 388,
"column": 77
} | {
"line": 388,
"column": 77
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : ¬u = v\nhadj : ¬G.Adj u v\nhacyc : (G ⊔ edge u v).IsAcyclic\nhreach : G.Reachable u v\n⊢ s(u, v) ∈ (G ⊔ edge u v).edgeSet",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"False",
"SimpleGraph.edge",
"eq_false",
... | [] | simp [huv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 388,
"column": 67
} | {
"line": 388,
"column": 77
} | {
"line": 388,
"column": 77
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : ¬u = v\nhadj : ¬G.Adj u v\nhacyc : (G ⊔ edge u v).IsAcyclic\nhreach : G.Reachable u v\n⊢ s(u, v) ∈ (G ⊔ edge u v).edgeSet",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"False",
"SimpleGraph.edge",
"eq_false",
... | [] | simp [huv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 388,
"column": 67
} | {
"line": 388,
"column": 77
} | {
"line": 388,
"column": 77
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : ¬u = v\nhadj : ¬G.Adj u v\nhacyc : (G ⊔ edge u v).IsAcyclic\nhreach : G.Reachable u v\n⊢ s(u, v) ∈ (G ⊔ edge u v).edgeSet",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"False",
"SimpleGraph.edge",
"eq_false",
... | [] | simp [huv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 419,
"column": 57
} | {
"line": 419,
"column": 68
} | {
"line": 419,
"column": 69
} | [
{
"pp": "V : Type u_1\nG F : SimpleGraph V\nhle : F ≤ G\nhF : F.IsAcyclic\nh : F.Reachable = G.Reachable\nthis : ¬Maximal (fun F ↦ F ≤ G ∧ F.IsAcyclic) F\nH : SimpleGraph V\nhFH : F < H\nhHG : H ≤ G\nhH : H.IsAcyclic\ne : Sym2 V\nheH : e ∈ H.edgeSet\nheF : e ∉ F.edgeSet\nh_bridge : (F ⊔ fromEdgeSet {e}).IsBridg... | [
"V : Type u_1\nG F : SimpleGraph V\nhle : F ≤ G\nhF : F.IsAcyclic\nh : F.Reachable = G.Reachable\nthis : ¬Maximal (fun F ↦ F ≤ G ∧ F.IsAcyclic) F\nH : SimpleGraph V\nhFH : F < H\nhHG : H ≤ G\nhH : H.IsAcyclic\ne : Sym2 V\nheH : e ∈ H.edgeSet\nheF : e ∉ F.edgeSet\nh_bridge : (F ⊔ fromEdgeSet {e}).IsBridge e\n⊢ e ∉ F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 487,
"column": 35
} | {
"line": 487,
"column": 59
} | {
"line": 487,
"column": 60
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nthis : Fintype V\nh : G.IsTree\n⊢ Nat.card ↑G.edgeSet + 1 = Nat.card V",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_ofFinset",
"SimpleGraph.decidableMemEdgeSet",
"Finset.univ",
... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nthis : Fintype V\nh : G.IsTree\n⊢ (Finset.filter (Membership.mem G.edgeSet) Finset.univ).card + 1 = Fintype.card V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 489,
"column": 2
} | {
"line": 489,
"column": 45
} | {
"line": 490,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nthis : Fintype V\nx✝ : G.Connected ∧ Nat.card ↑G.edgeSet + 1 = Nat.card V\nh₁ : G.Connected\nh₂ : Nat.card ↑G.edgeSet + 1 = Nat.card V\n⊢ G.IsAcyclic",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym2.mk",... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nthis : Fintype V\nx✝ : G.Connected ∧ Nat.card ↑G.edgeSet + 1 = Nat.card V\nh₁ : G.Connected\nh₂ : Nat.card ↑G.edgeSet + 1 = Nat.card V\n⊢ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w)"
] | simp_rw [isAcyclic_iff_forall_adj_isBridge] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Coloring.EdgeLabeling | {
"line": 202,
"column": 56
} | {
"line": 202,
"column": 79
} | {
"line": 202,
"column": 80
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nx y : V\n⊢ (EdgeLabeling.labelGraph G.toTopEdgeLabeling 1).Adj x y ↔ G.Adj x y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"SimpleGraph.Adj",
"Exists",
... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nx y : V\n⊢ G.Adj x y → ¬x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 214,
"column": 31
} | {
"line": 214,
"column": 42
} | {
"line": 214,
"column": 43
} | [
{
"pp": "V : Type u_1\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nr : ℕ\nh : G.IsTuranMaximal r\ninst✝ : DecidableEq V\nfp : Finpartition univ := h.finpartition\nlarge : Finset V\nhl : large ∈ fp.parts\nsmall : Finset V\nhs : small ∈ fp.parts\nineq : #small + 1 < #large\nw : V\nhw : w ∈... | [
"V : Type u_1\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nr : ℕ\nh : G.IsTuranMaximal r\ninst✝ : DecidableEq V\nfp : Finpartition univ := h.finpartition\nlarge : Finset V\nhl : large ∈ fp.parts\nsmall : Finset V\nhs : small ∈ fp.parts\nineq : #small + 1 < #large\nw : V\nhw : w ∈ large\nv : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite | {
"line": 79,
"column": 28
} | {
"line": 79,
"column": 39
} | {
"line": 79,
"column": 40
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝⁴ : DecidableEq V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\nG' : SimpleGraph V\nh : G ≤ G'\nc' : G'.ConnectedComponent\ninst✝¹ : Fintype ↑c'.supp\ninst✝ : DecidablePred fun c ↦ c.supp ⊆ c'.supp\nx : G.ConnectedComponent\nx✝¹ : x ∈ ↑{c | c.supp ⊆ c'.supp}\ny :... | [
"V : Type u\nG : SimpleGraph V\ninst✝⁴ : DecidableEq V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\nG' : SimpleGraph V\nh : G ≤ G'\nc' : G'.ConnectedComponent\ninst✝¹ : Fintype ↑c'.supp\ninst✝ : DecidablePred fun c ↦ c.supp ⊆ c'.supp\nx : G.ConnectedComponent\nx✝¹ : x ∈ ↑{c | c.supp ⊆ c'.supp}\ny : G.Connected... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite | {
"line": 81,
"column": 29
} | {
"line": 81,
"column": 40
} | {
"line": 81,
"column": 41
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝⁴ : DecidableEq V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\nG' : SimpleGraph V\nh : G ≤ G'\nc' : G'.ConnectedComponent\ninst✝¹ : Fintype ↑c'.supp\ninst✝ : DecidablePred fun c ↦ c.supp ⊆ c'.supp\n⊢ ↑({c | c.supp ⊆ c'.supp}.disjiUnion (fun c ↦ c.supp.toFinset) ... | [
"V : Type u\nG : SimpleGraph V\ninst✝⁴ : DecidableEq V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\nG' : SimpleGraph V\nh : G ≤ G'\nc' : G'.ConnectedComponent\ninst✝¹ : Fintype ↑c'.supp\ninst✝ : DecidablePred fun c ↦ c.supp ⊆ c'.supp\n⊢ ⋃ x, ⋃ (_ : x.supp ⊆ c'.supp), x.supp = c'.supp"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 33
} | {
"line": 331,
"column": 34
} | [
{
"pp": "α : Type u\nG : SimpleGraph α\ninst✝ : Fintype α\nn : ℕ\nC : G.Coloring (Fin n)\nt : ℕ\nh : ∀ (c : Fin n), Fintype.card ↑(C.colorClass c) ≤ t\nthis : ∀ (c : Fin n), Nonempty (↑(C.colorClass c) ↪ Fin t)\nF : (c : Fin n) → ↑(C.colorClass c) ↪ Fin t\nc₁ c₂ : Fin n\nv₁ : ↑(C.colorClass c₁)\nv₂ : ↑(C.colorC... | [
"α : Type u\nG : SimpleGraph α\ninst✝ : Fintype α\nn : ℕ\nC : G.Coloring (Fin n)\nt : ℕ\nh : ∀ (c : Fin n), Fintype.card ↑(C.colorClass c) ≤ t\nthis : ∀ (c : Fin n), Nonempty (↑(C.colorClass c) ↪ Fin t)\nF : (c : Fin n) → ↑(C.colorClass c) ↪ Fin t\nc₁ c₂ : Fin n\nv₁ : ↑(C.colorClass c₁)\nv₂ : ↑(C.colorClass c₂)\nhc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 385,
"column": 18
} | {
"line": 385,
"column": 34
} | {
"line": 385,
"column": 35
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\n⊢ #(K.parts.disjiUnion id ⋯) = r * t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"SimpleGraph.CompleteEquipartiteSubgraph.disjoint",
... | [
"V : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\n⊢ ∑ a ∈ K.parts, #(id a) = r * t"
] | card_disjiUnion, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 42,
"column": 4
} | {
"line": 44,
"column": 9
} | {
"line": 44,
"column": 9
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nC : Set G.ConnectedComponent\n⊢ Set.InjOn G.connectedComponentMk (Quot.out '' C)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.connectedComponentMk",
"Quot.out",
"congrArg",
"Quot.out_eq",
"Membership.mem"... | [] | rintro x ⟨c, ⟨hc, rfl⟩⟩ y ⟨d, ⟨hd, rfl⟩⟩ hxy
simp only [connectedComponentMk] at hxy
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 42,
"column": 4
} | {
"line": 44,
"column": 9
} | {
"line": 44,
"column": 9
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nC : Set G.ConnectedComponent\n⊢ Set.InjOn G.connectedComponentMk (Quot.out '' C)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.connectedComponentMk",
"Quot.out",
"congrArg",
"Quot.out_eq",
"Membership.mem"... | [] | rintro x ⟨c, ⟨hc, rfl⟩⟩ y ⟨d, ⟨hd, rfl⟩⟩ hxy
simp only [connectedComponentMk] at hxy
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 279,
"column": 2
} | {
"line": 280,
"column": 60
} | {
"line": 282,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn r : ℕ\nf : G ≃g turanGraph n r\nhr : 0 < r\nJ : SimpleGraph V\nw✝ : DecidableRel J.Adj\nj : J.IsTuranMaximal r\ng : J ≃g turanGraph n r\n⊢ G.IsTuranMaximal r",
"ppTerm": "?m.62",
"assigned": true,
"usedConsta... | [] | use (turanGraph_cliqueFree (n := n) hr).comap f.isContained,
fun H _ cf ↦ (f.symm.comp g).card_edgeFinset_eq ▸ j.2 cf | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 87,
"column": 10
} | {
"line": 87,
"column": 76
} | {
"line": 87,
"column": 77
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : Even K.supp.ncard\n⊢ ?m.31 ∉ G.oddComponents",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"SimpleGraph.oddComponents",
"Membership.mem",
"id",
"S... | [
"V : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : Even K.supp.ncard\n⊢ ¬Odd (supp ?m.31).ncard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 86,
"column": 4
} | {
"line": 87,
"column": 86
} | {
"line": 87,
"column": 87
} | [
{
"pp": "case pos\nV : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : Even K.supp.ncard\n⊢ Even (K.supp \\ s).ncard",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"_private.Mathlib.Combinator... | [
"case pos\nV : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : Even K.supp.ncard\n⊢ Even K.supp.ncard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 30
} | {
"line": 90,
"column": 31
} | [
{
"pp": "case neg\nV : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : ¬Even K.supp.ncard\nthis : K.supp.ncard ≠ 0\n⊢ Even K.supp.ncard ↔ Even 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Nat.not_even_iff_odd._simp_1",
... | [
"case neg\nV : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : ¬Even K.supp.ncard\nthis : K.supp.ncard ≠ 0\n⊢ Odd K.supp.ncard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 41
} | {
"line": 63,
"column": 42
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Subsingleton α\nu : α\n⊢ G.eccent u = 0",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"SimpleGraph.edist_eq_zero_iff._simp_1",
"Eq.mpr",
"instCompleteLinearOrderENat",
"CommSemiring.toSemiring",
"iSup",
... | [
"α : Type u_1\nG : SimpleGraph α\ninst✝ : Subsingleton α\nu : α\n⊢ ∀ (i : α), u = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 424,
"column": 6
} | {
"line": 424,
"column": 17
} | {
"line": 424,
"column": 18
} | [
{
"pp": "case neg.refine_1\nα : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nf : (completeEquipartiteGraph r t).Copy G\nht : ¬t = 0\ni : Fin r\nx✝¹ x✝ : Fin t\nh : (fun j ↦ f (i, j)) x✝¹ = (fun j ↦ f (i, j)) x✝\n⊢ x✝¹ = x✝",
"ppTerm"... | [
"case neg.refine_1\nα : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nf : (completeEquipartiteGraph r t).Copy G\nht : ¬t = 0\ni : Fin r\nx✝¹ x✝ : Fin t\nh : (fun j ↦ f (i, j)) x✝¹ = (fun j ↦ f (i, j)) x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 427,
"column": 57
} | {
"line": 427,
"column": 68
} | {
"line": 427,
"column": 69
} | [
{
"pp": "α : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nf : (completeEquipartiteGraph r t).Copy G\nht : ¬t = 0\ni₁ i₂ : Fin r\nh :\n ∀ (a : V),\n a ∈ map { toFun := fun j ↦ f (i₁, j), inj' := ⋯ } univ ↔ a ∈ map { toFun := fun j ↦ f... | [
"α : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nf : (completeEquipartiteGraph r t).Copy G\nht : ¬t = 0\ni₁ i₂ : Fin r\nh :\n ∀ (a : V),\n a ∈ map { toFun := fun j ↦ f (i₁, j), inj' := ⋯ } univ ↔ a ∈ map { toFun := fun j ↦ f (i₂, j), in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 377,
"column": 2
} | {
"line": 378,
"column": 17
} | {
"line": 379,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : SimpleGraph α\nh : G.radius = 0\n⊢ Nonempty α",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"iInf",
"instCompleteLinearOrderENat",
"congrArg",
"CommSemiring.toSemiring",
"ENat.iInf_... | [
"case refine_2\nα : Type u_1\nG : SimpleGraph α\nh : G.radius = 0\n⊢ Subsingleton α",
"case refine_3\nα : Type u_1\nG : SimpleGraph α\nx✝ : Nonempty α ∧ Subsingleton α\nleft✝ : Nonempty α\nright✝ : Subsingleton α\n⊢ G.radius = 0"
] | · contrapose! h
simp [radius] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 24
} | {
"line": 383,
"column": 25
} | [
{
"pp": "case h\nα : Type u_1\nG : SimpleGraph α\nx✝ : Nonempty α ∧ Subsingleton α\nleft✝ : Nonempty α\nright✝ : Subsingleton α\n⊢ G.eccent Classical.ofNonempty = 0",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"SimpleGraph.edist_eq_zero_iff._simp_1",
"Eq.mpr",
"Classical.... | [
"case h\nα : Type u_1\nG : SimpleGraph α\nx✝ : Nonempty α ∧ Subsingleton α\nleft✝ : Nonempty α\nright✝ : Subsingleton α\n⊢ ∀ (i : α), Classical.ofNonempty = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Finsubgraph | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 12
} | {
"line": 163,
"column": 4
} | [
{
"pp": "V : Type u\nW : Type v\nG : SimpleGraph V\nF : SimpleGraph W\ninst✝ : Finite W\nh : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F\nval✝ : Fintype W\nthis : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')\n⊢ (G' : G.Finsubgraphᵒᵖ) → Fintype ((G.finsubgraphHomFunctor F).obj G'... | [
"V : Type u\nW : Type v\nG : SimpleGraph V\nF : SimpleGraph W\ninst✝ : Finite W\nh : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F\nval✝ : Fintype W\nthis : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')\nG' : G.Finsubgraphᵒᵖ\n⊢ Fintype ((G.finsubgraphHomFunctor F).obj G')"
] | intro G' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.SimpleGraph.Finsubgraph | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 12
} | {
"line": 163,
"column": 4
} | [
{
"pp": "V : Type u\nW : Type v\nG : SimpleGraph V\nF : SimpleGraph W\ninst✝ : Finite W\nh : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F\nval✝ : Fintype W\nthis : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')\n⊢ (G' : G.Finsubgraphᵒᵖ) → Fintype ((G.finsubgraphHomFunctor F).obj G'... | [
"V : Type u\nW : Type v\nG : SimpleGraph V\nF : SimpleGraph W\ninst✝ : Finite W\nh : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F\nval✝ : Fintype W\nthis : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')\nG' : G.Finsubgraphᵒᵖ\n⊢ Fintype ((G.finsubgraphHomFunctor F).obj G')"
] | intro G' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 45
} | {
"line": 130,
"column": 2
} | [
{
"pp": "W : Type u_1\nH : SimpleGraph W\nε : ℝ\nhε_pos : 0 < ε\nh : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ∃ G inst, ↑(#G.edgeFinset) ≥ (H.turanDensity + ε) * ↑(b.choose 2) ∧ IsEmpty (H.Copy G)\n⊢ H.turanDensity + ε ≤ sInf {x | ∃ n ∈ Set.Ici 2, ↑(extremalNumber n H) / ↑(n.choose 2) = x}",
"ppTerm": "?m.61",
"assigned... | [
"case refine_1\nW : Type u_1\nH : SimpleGraph W\nε : ℝ\nhε_pos : 0 < ε\nh : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ∃ G inst, ↑(#G.edgeFinset) ≥ (H.turanDensity + ε) * ↑(b.choose 2) ∧ IsEmpty (H.Copy G)\n⊢ {x | ∃ n ∈ Set.Ici 2, ↑(extremalNumber n H) / ↑(n.choose 2) = x}.Nonempty",
"case refine_2\nW : Type u_1\nH : SimpleGraph W\... | refine le_csInf ?_ (fun x ⟨m, hm, hx⟩ ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 78,
"column": 25
} | {
"line": 78,
"column": 41
} | {
"line": 78,
"column": 42
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nv : Fin n\nht'_pos : 0 < t'\nhv : v ∈ K.vertsᶜ\np : Finset (Fin n)\nhp : p ∈ K.parts\nhs : ∀ (x : Finset (Fin n)), x ∉ powersetCard t p ∨ ∃ x_1 ∈ x, ¬G.Adj v x_1\n⊢ #(K.parts.disjiUnion (fun ... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nv : Fin n\nht'_pos : 0 < t'\nhv : v ∈ K.vertsᶜ\np : Finset (Fin n)\nhp : p ∈ K.parts\nhs : ∀ (x : Finset (Fin n)), x ∉ powersetCard t p ∨ ∃ x_1 ∈ x, ¬G.Adj v x_1\n⊢ ∑ a ∈ K.parts, #({v_1 ∈ id a | (betwee... | card_disjiUnion, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 59
} | {
"line": 166,
"column": 60
} | [
{
"pp": "W : Type u_1\nH : SimpleGraph W\nε : ℝ\nhε_pos : 0 < ε\nV : Type u_2\ninst✝¹ : Fintype V\nh_verts : Fintype.card V ≥ H.turanDensityConst ε\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\n⊢ Nat.find ⋯ ≤ Fintype.card V",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"W : Type u_1\nH : SimpleGraph W\nε : ℝ\nhε_pos : 0 < ε\nV : Type u_2\ninst✝¹ : Fintype V\nh_verts : Fintype.card V ≥ H.turanDensityConst ε\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\n⊢ Nat.find ⋯ ≤ Fintype.card V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Girth | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | {
"line": 76,
"column": 14
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\n⊢ 3 ≤ G.egirth",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"SimpleGraph.le_egirth._simp_1",
"Eq.mpr",
"instCompleteLinearOrderENat",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ChainCompletePartialOrder.i... | [
"α : Type u_1\nG : SimpleGraph α\n⊢ ∀ (a : α) (w : G.Walk a a), w.IsCycle → 3 ≤ w.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 111,
"column": 10
} | {
"line": 112,
"column": 78
} | {
"line": 113,
"column": 4
} | [
{
"pp": "case h₂\nn : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nv : Fin n\nhv : v ∈ filter K t\n⊢ ↑((between (↑K.verts) (↑K.verts)ᶜ G).degree v) ≤ ↑(#K.verts)",
"ppTerm": "?h₂",
"assigned": true,
"use... | [] | exact_mod_cast isBipartiteWith_degree_le'
(between_verts_isBipartiteWith K) (filter_subset_compl_verts K hv) | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacticExact_mod_cast__1 | Lean.Parser.Tactic.tacticExact_mod_cast_ |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 111,
"column": 10
} | {
"line": 112,
"column": 78
} | {
"line": 113,
"column": 4
} | [
{
"pp": "case h₂\nn : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nv : Fin n\nhv : v ∈ filter K t\n⊢ ↑((between (↑K.verts) (↑K.verts)ᶜ G).degree v) ≤ ↑(#K.verts)",
"ppTerm": "?h₂",
"assigned": true,
"use... | [] | exact_mod_cast isBipartiteWith_degree_le'
(between_verts_isBipartiteWith K) (filter_subset_compl_verts K hv) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 111,
"column": 10
} | {
"line": 112,
"column": 78
} | {
"line": 113,
"column": 4
} | [
{
"pp": "case h₂\nn : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nv : Fin n\nhv : v ∈ filter K t\n⊢ ↑((between (↑K.verts) (↑K.verts)ᶜ G).degree v) ≤ ↑(#K.verts)",
"ppTerm": "?h₂",
"assigned": true,
"use... | [] | exact_mod_cast isBipartiteWith_degree_le'
(between_verts_isBipartiteWith K) (filter_subset_compl_verts K hv) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 51
} | {
"line": 137,
"column": 52
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\n⊢ (⋃ i ∈ t, s i).ncard ≤ ∑ᶠ (i : ι) (_ : i ∈ t), (s i).ncard",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\n⊢ (⋃ i ∈ t, s i).ncard ≤ ∑ᶠ (i : ι) (_ : i ∈ t), (s i).ncard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 51
} | {
"line": 141,
"column": 52
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\n⊢ (⋃ i ∈ t, s i).encard ≤ ∑ᶠ (i : ι) (_ : i ∈ t), (s i).encard",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\n⊢ (⋃ i ∈ t, s i).encard ≤ ∑ᶠ (i : ι) (_ : i ∈ t), (s i).encard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 13
} | {
"line": 145,
"column": 14
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Fintype ι\ns : ι → Set α\n⊢ (⋃ i, s i).ncard ≤ ∑ i, (s i).ncard",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nι : Type u_2\ninst✝ : Fintype ι\ns : ι → Set α\n⊢ (⋃ i, s i).ncard ≤ ∑ i, (s i).ncard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | {
"line": 149,
"column": 14
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Fintype ι\ns : ι → Set α\n⊢ (⋃ i, s i).encard ≤ ∑ i, (s i).encard",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nι : Type u_2\ninst✝ : Fintype ι\ns : ι → Set α\n⊢ (⋃ i, s i).encard ≤ ∑ i, (s i).encard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 13
} | {
"line": 153,
"column": 14
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Finite ι\ns : ι → Set α\n⊢ (⋃ i, s i).ncard ≤ ∑ᶠ (i : ι), (s i).ncard",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nι : Type u_2\ninst✝ : Finite ι\ns : ι → Set α\n⊢ (⋃ i, s i).ncard ≤ ∑ᶠ (i : ι), (s i).ncard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 13
} | {
"line": 157,
"column": 14
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Finite ι\ns : ι → Set α\n⊢ (⋃ i, s i).encard ≤ ∑ᶠ (i : ι), (s i).encard",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nι : Type u_2\ninst✝ : Finite ι\ns : ι → Set α\n⊢ (⋃ i, s i).encard ≤ ∑ᶠ (i : ι), (s i).encard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 18
} | {
"line": 50,
"column": 4
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\np : Set V\ninst✝ : DecidablePred fun x ↦ x ∈ p\nf : ↑p → V\nh₁ : ∀ (x : ↑p), f x ∉ p\nh₂ : ∀ (x : ↑p), G.Adj (↑x) (f x)\nv w : V\nh : if h : v ∈ p then f ⟨v, h⟩ = w else if h : w ∈ p then f ⟨w, h⟩ = v else False\n⊢ G.Adj v w",
"ppTerm": "?m.47",
"assigned": true... | [
"case pos\nV : Type u_1\nG : SimpleGraph V\np : Set V\ninst✝ : DecidablePred fun x ↦ x ∈ p\nf : ↑p → V\nh₁ : ∀ (x : ↑p), f x ∉ p\nh₂ : ∀ (x : ↑p), G.Adj (↑x) (f x)\nv w : V\nh✝ : v ∈ p\nh : f ⟨v, h✝⟩ = w\n⊢ G.Adj v w",
"case pos\nV : Type u_1\nG : SimpleGraph V\np : Set V\ninst✝ : DecidablePred fun x ↦ x ∈ p\nf :... | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 178,
"column": 8
} | {
"line": 178,
"column": 51
} | {
"line": 179,
"column": 4
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nht_lt_t' : t < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nhN : (↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\nthis ... | [] | simp_rw [Finset.card_pi, card_powersetCard] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 178,
"column": 8
} | {
"line": 178,
"column": 51
} | {
"line": 179,
"column": 4
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nht_lt_t' : t < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nhN : (↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\nthis ... | [] | simp_rw [Finset.card_pi, card_powersetCard] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 178,
"column": 8
} | {
"line": 178,
"column": 51
} | {
"line": 179,
"column": 4
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nht_lt_t' : t < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nhN : (↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\nthis ... | [] | simp_rw [Finset.card_pi, card_powersetCard] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 13
} | {
"line": 106,
"column": 14
} | [
{
"pp": "case h\nV : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nh : M.IsMatching\nhGG' : G ≤ G'\nv✝ : V\nhv✝ : v✝ ∈ (Subgraph.map (Hom.ofLE hGG') M).verts\nw✝ : V\nhv : w✝ ∈ M.verts\nhv' : (Hom.ofLE hGG') w✝ = v✝\nw : V\nhw : (fun w ↦ M.Adj w✝ w) w ∧ ∀ (y : V), (fun w ↦ M.Adj w✝ w) y → y = w\n⊢ (fun w ↦ (S... | [
"case h\nV : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nh : M.IsMatching\nhGG' : G ≤ G'\nv✝ : V\nhv✝ : v✝ ∈ (Subgraph.map (Hom.ofLE hGG') M).verts\nw✝ : V\nhv : w✝ ∈ M.verts\nhv' : (Hom.ofLE hGG') w✝ = v✝\nw : V\nhw : (fun w ↦ M.Adj w✝ w) w ∧ ∀ (y : V), (fun w ↦ M.Adj w✝ w) y → y = w\n⊢ M.Adj v✝ w ∧ ∀ (y : V),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 77
} | {
"line": 133,
"column": 78
} | [
{
"pp": "case h.inr\nV : Type u_1\nG : SimpleGraph V\nM M' : G.Subgraph\nhM : M.IsMatching\nhM' : M'.IsMatching\nhd✝ : Disjoint M.support M'.support\nv : V\nhv : v ∈ (M ⊔ M').verts\nN N' : G.Subgraph\nhN : N.IsMatching\nhd : ∀ ⦃a : V⦄, a ∈ N.support → a ∉ N'.support\nhmN : v ∈ N.verts\nw : V\nhw : (fun w ↦ N.Ad... | [
"case h.inr\nV : Type u_1\nG : SimpleGraph V\nM M' : G.Subgraph\nhM : M.IsMatching\nhM' : M'.IsMatching\nhd✝ : Disjoint M.support M'.support\nv : V\nhv : v ∈ (M ⊔ M').verts\nN N' : G.Subgraph\nhN : N.IsMatching\nhd : ∀ ⦃a : V⦄, a ∈ N.support → a ∉ N'.support\nhmN : v ∈ N.verts\nw : V\nhw : (fun w ↦ N.Adj v w) w ∧ ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 77
} | {
"line": 154,
"column": 78
} | [
{
"pp": "case neg\nV : Type u_1\nG : SimpleGraph V\nι : Type u_3\nf : ι → G.Subgraph\nhM : ∀ (i : ι), (f i).IsMatching\nhd : Pairwise fun i j ↦ Disjoint (f i).support (f j).support\nv : V\nhv : v ∈ (⨆ i, f i).verts\ni : ι\nhi : v ∈ (f i).verts\nw : V\nhw : (fun w ↦ (f i).Adj v w) w ∧ ∀ (y : V), (fun w ↦ (f i).A... | [
"case neg\nV : Type u_1\nG : SimpleGraph V\nι : Type u_3\nf : ι → G.Subgraph\nhM : ∀ (i : ι), (f i).IsMatching\nhd : Pairwise fun i j ↦ Disjoint (f i).support (f j).support\nv : V\nhv : v ∈ (⨆ i, f i).verts\ni : ι\nhi : v ∈ (f i).verts\nw : V\nhw : (fun w ↦ (f i).Adj v w) w ∧ ∀ (y : V), (fun w ↦ (f i).Adj v w) y → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 146,
"column": 11
} | {
"line": 146,
"column": 26
} | {
"line": 146,
"column": 27
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na b : α\np : G.Walk a b\ninst✝ : Fintype α\nh✝ : Nonempty α\nx✝ : p.IsPath ∧ p.length = Fintype.card α - 1\nhp : p.IsPath\nh : p.length = Fintype.card α - 1\nthis : Injective fun x ↦ p.support.get x\n⊢ Fintype.card α = p.support.length"... | [
"case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na b : α\np : G.Walk a b\ninst✝ : Fintype α\nh✝ : Nonempty α\nx✝ : p.IsPath ∧ p.length = Fintype.card α - 1\nhp : p.IsPath\nh : p.length = Fintype.card α - 1\nthis : Injective fun x ↦ p.support.get x\n⊢ Fintype.card α = p.length + 1"
] | length_support, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 15
} | {
"line": 167,
"column": 16
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\nβ : Type u_2\ninst✝ : DecidableEq β\nH : SimpleGraph β\na : α\nf : G →g H\nhf : Bijective ⇑f\nx v✝ : α\ny : G.Adj a v✝\np : G.Walk v✝ a\nhp : (cons y p).IsHamiltonianCycle\n__IsCycle✝ : (Walk.map f (cons y p)).IsCycle := IsCycle.map (B... | [
"case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\nβ : Type u_2\ninst✝ : DecidableEq β\nH : SimpleGraph β\na : α\nf : G →g H\nhf : Bijective ⇑f\nx v✝ : α\ny : G.Adj a v✝\np : G.Walk v✝ a\nhp : (cons y p).IsHamiltonianCycle\n__IsCycle✝ : (Walk.map f (cons y p)).IsCycle := IsCycle.map (Bijective.inj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 194,
"column": 2
} | {
"line": 196,
"column": 11
} | {
"line": 198,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na : α\np : G.Walk a a\ninst✝ : Fintype α\nhp : p.IsHamiltonianCycle\n⊢ p.length = Fintype.card α",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Walk.length_tail_add_one",
"SimpleGrap... | [] | rw [← length_tail_add_one hp.not_nil, hp.isHamiltonian_tail.length_eq, Nat.sub_add_cancel]
rw [Nat.succ_le_iff, Fintype.card_pos_iff]
exact ⟨a⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 194,
"column": 2
} | {
"line": 196,
"column": 11
} | {
"line": 198,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na : α\np : G.Walk a a\ninst✝ : Fintype α\nhp : p.IsHamiltonianCycle\n⊢ p.length = Fintype.card α",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Walk.length_tail_add_one",
"SimpleGrap... | [] | rw [← length_tail_add_one hp.not_nil, hp.isHamiltonian_tail.length_eq, Nat.sub_add_cancel]
rw [Nat.succ_le_iff, Fintype.card_pos_iff]
exact ⟨a⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 215,
"column": 13
} | {
"line": 215,
"column": 37
} | {
"line": 215,
"column": 38
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nG' : SimpleGraph W\nM : G.Subgraph\nf : G ≃g G'\nh : (Subgraph.map f.toHom M).IsMatching\n⊢ M.IsMatching",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nG' : SimpleGraph W\nM : G.Subgraph\nf : G ≃g G'\nh : (Subgraph.map f.toHom M).IsMatching\n⊢ M.IsMatching"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 35
} | {
"line": 268,
"column": 36
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Fintype V\nh : M.IsPerfectMatching\n⊢ Even (Fintype.card V)",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Fintype V\nh : M.IsPerfectMatching\n⊢ Even (Fintype.card V)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 77
} | {
"line": 276,
"column": 78
} | [
{
"pp": "case h\nV : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\nh : M.IsMatching\nv✝ : V\nhv : v✝ ∈ M.verts\nw : V\nhvw : M.Adj v✝ w\nhw : ∀ (y : V), (fun w ↦ M.Adj v✝ w) y → y = w\n⊢ (fun w ↦ (M.induce (M.verts ∩ (G.connectedComponentMk v✝).supp)).Adj v✝ w) w ∧\n ∀ (y : V), (fun w ↦ (M.induce (M.verts ∩ (... | [
"case h\nV : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\nh : M.IsMatching\nv✝ : V\nhv : v✝ ∈ M.verts\nw : V\nhvw : M.Adj v✝ w\nhw : ∀ (y : V), (fun w ↦ M.Adj v✝ w) y → y = w\n⊢ ∀ y ∈ M.verts, G.Reachable y v✝ → M.Adj v✝ y → y = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 33
} | {
"line": 280,
"column": 34
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\nh : M.IsPerfectMatching\nc : G.ConnectedComponent\n⊢ (M.induce c.supp).IsMatching",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\nh : M.IsPerfectMatching\nc : G.ConnectedComponent\n⊢ (M.induce c.supp).IsMatching"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 46
} | {
"line": 340,
"column": 47
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Finite V\nu : Set V\nhM : M.IsPerfectMatching\nc : ↑(⊤.deleteVerts u).coe.oddComponents\nh : ∀ w ∈ u, ∀ (v : ↑(⊤.deleteVerts u).verts), M.Adj (↑v) w → v ∉ (↑c).supp\nhMmatch : (M.induce (Subtype.val '' (↑c).supp)).IsMatching\nthis✝ : Fintype ↑(M.... | [
"V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Finite V\nu : Set V\nhM : M.IsPerfectMatching\nc : ↑(⊤.deleteVerts u).coe.oddComponents\nh : ∀ w ∈ u, ∀ (v : ↑(⊤.deleteVerts u).verts), M.Adj (↑v) w → v ∉ (↑c).supp\nhMmatch : (M.induce (Subtype.val '' (↑c).supp)).IsMatching\nthis✝ : Fintype ↑(M.induce (Subt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 364,
"column": 10
} | {
"line": 364,
"column": 25
} | {
"line": 364,
"column": 26
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\n... | [
"α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\nW : Finset α... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 21
} | {
"line": 380,
"column": 22
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\n... | [
"α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\nW : Finset α... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 13
} | {
"line": 132,
"column": 14
} | [
{
"pp": "V : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nv w : V\nh : G.Adj v w\n⊢ ((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \\ ({w} ∪ {v}) = (G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ",
"ppTerm": "?m.36",
"assigned": true,
"usedConstan... | [
"V : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nv w : V\nh : G.Adj v w\n⊢ G.Adj v w ∧ G.Adj w v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 152,
"column": 27
} | {
"line": 152,
"column": 49
} | {
"line": 152,
"column": 50
} | [
{
"pp": "case inl\nV : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝ : DecidableEq V\nh : G.IsSRGWith n k ℓ μ\nv u : V\nha : v ≠ u ∧ ¬G.Adj v u\nhne : v ≠ u\n⊢ ¬G.Adj v u ∧ ¬G.Adj u u",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.... | [
"case inl\nV : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝ : DecidableEq V\nh : G.IsSRGWith n k ℓ μ\nv u : V\nha : v ≠ u ∧ ¬G.Adj v u\nhne : v ≠ u\n⊢ ¬G.Adj v u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 152,
"column": 27
} | {
"line": 152,
"column": 49
} | {
"line": 152,
"column": 50
} | [
{
"pp": "case inr\nV : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝ : DecidableEq V\nh : G.IsSRGWith n k ℓ μ\nw u : V\nha : u ≠ w ∧ ¬G.Adj u w\nhne : u ≠ w\n⊢ ¬G.Adj u u ∧ ¬G.Adj w u",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.... | [
"case inr\nV : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝ : DecidableEq V\nh : G.IsSRGWith n k ℓ μ\nw u : V\nha : u ≠ w ∧ ¬G.Adj u w\nhne : u ≠ w\n⊢ ¬G.Adj w u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 609,
"column": 47
} | {
"line": 609,
"column": 58
} | {
"line": 609,
"column": 59
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : G'.Adj v w\nw' : V\nhw' : w ≠ w' ∧ G'.Adj v w'\n⊢ M.Adj v w ↔ ¬M.Adj v w'",
"... | [
"V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : G'.Adj v w\nw' : V\nhw' : w ≠ w' ∧ G'.Adj v w'\n⊢ M.Adj v w ↔ ¬M.Adj v w'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 617,
"column": 48
} | {
"line": 617,
"column": 59
} | {
"line": 617,
"column": 60
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : G'.Adj v w\nw' : V\nhw' : w ≠ w' ∧ G'.Adj v w'\nhmadj : M.Adj v w ↔ ¬M.Adj v w'\n... | [
"V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : G'.Adj v w\nw' : V\nhw' : w ≠ w' ∧ G'.Adj v w'\nhmadj : M.Adj v w ↔ ¬M.Adj v w'\ny : V\nhr : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 624,
"column": 48
} | {
"line": 624,
"column": 59
} | {
"line": 624,
"column": 60
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : ¬G'.Adj v w\ny : V\nhr : G'.Adj v y ∧ ¬M.Adj v y\nw' : V\nhw' : y ≠ w' ∧ G'.Adj v... | [
"V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : ¬G'.Adj v w\ny : V\nhr : G'.Adj v y ∧ ¬M.Adj v y\nw' : V\nhw' : y ≠ w' ∧ G'.Adj v w'\n⊢ M.Adj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 29
} | {
"line": 638,
"column": 30
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nM' : G'.Subgraph\nhM : M.IsPerfectMatching\nhM' : M'.IsPerfectMatching\n⊢ (M.spanningCoe ∆ M'.spanningCoe).IsAlternating M'.spanningCoe",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nM' : G'.Subgraph\nhM : M.IsPerfectMatching\nhM' : M'.IsPerfectMatching\n⊢ (M.spanningCoe ∆ M'.spanningCoe).IsAlternating M'.spanningCoe"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 214,
"column": 2
} | {
"line": 215,
"column": 31
} | {
"line": 216,
"column": 2
} | [
{
"pp": "V : Type u\ninst✝³ : Fintype V\nG : SimpleGraph V\ninst✝² : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝¹ : DecidableEq V\nα : Type u_1\ninst✝ : Semiring α\nh : G.IsSRGWith n k ℓ μ\nv w : V\n⊢ (adjMatrix α G ^ 2) v w = (k • 1 + ℓ • adjMatrix α G + μ • adjMatrix α Gᶜ) v w",
"ppTerm": "?m.64",
"assigne... | [
"V : Type u\ninst✝³ : Fintype V\nG : SimpleGraph V\ninst✝² : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝¹ : DecidableEq V\nα : Type u_1\ninst✝ : Semiring α\nh : G.IsSRGWith n k ℓ μ\nv w : V\n⊢ ↑(Fintype.card ↑{p | p.length = 2}) =\n (k • 1 v w + ℓ • if G.Adj v w then 1 else 0) + μ • if v ≠ w ∧ ¬G.Adj v w then 1 else ... | simp only [adjMatrix_pow_apply_eq_card_walk, Matrix.add_apply, Matrix.smul_apply,
adjMatrix_apply, compl_adj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Trails | {
"line": 92,
"column": 16
} | {
"line": 92,
"column": 45
} | {
"line": 92,
"column": 46
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : G.Walk u v\nh : p.IsEulerian\ne : Sym2 V\nhe : e ∈ G.edgeSet\n⊢ e ∈ p.edges",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : G.Walk u v\nh : p.IsEulerian\ne : Sym2 V\nhe : e ∈ G.edgeSet\n⊢ e ∈ p.edges"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.UniversalVerts | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 78
} | {
"line": 62,
"column": 4
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns : Set ↑G.deleteUniversalVerts.verts\n⊢ Disjoint (Subtype.val '' s) G.universalVerts",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nG : SimpleGraph V\ns : Set ↑G.deleteUniversalVerts.verts\n⊢ Disjoint (Subtype.val '' s) G.universalVerts"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 400,
"column": 12
} | {
"line": 400,
"column": 52
} | {
"line": 401,
"column": 12
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\n... | [
"α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\nW : Finset α... | rw [← hw.card_inter, card_eq_zero] at hk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 79,
"column": 4
} | {
"line": 80,
"column": 11
} | {
"line": 80,
"column": 12
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nf : G ≃g H\nc : Set W\nh : G.IsVertexCover (⇑f ⁻¹' c)\n⊢ H.IsVertexCover c",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nf : G ≃g H\nc : Set W\nh : G.IsVertexCover (⇑f ⁻¹' c)\n⊢ H.IsVertexCover c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 17
} | {
"line": 128,
"column": 18
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nh : G.vertexCoverNum = 0\n⊢ G = ⊥",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nG : SimpleGraph V\nh : G.vertexCoverNum = 0\n⊢ G = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 135,
"column": 34
} | {
"line": 135,
"column": 64
} | {
"line": 135,
"column": 64
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\n⊢ G.IsVertexCover (Set.univ \\ {x})",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.... | [] | grind [IsVertexCover, Adj.ne'] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 135,
"column": 34
} | {
"line": 135,
"column": 64
} | {
"line": 135,
"column": 64
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\n⊢ G.IsVertexCover (Set.univ \\ {x})",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.... | [] | grind [IsVertexCover, Adj.ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 135,
"column": 34
} | {
"line": 135,
"column": 64
} | {
"line": 135,
"column": 64
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\n⊢ G.IsVertexCover (Set.univ \\ {x})",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.... | [] | grind [IsVertexCover, Adj.ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 66
} | {
"line": 136,
"column": 67
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\nthis : ↑n ≤ (Set.univ \\ {x}).encard\n⊢ ↑n ≤ ENat.card V - 1",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instAddMonoidWithOneENat",
... | [
"V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\nthis : ↑n ≤ (Set.univ \\ {x}).encard\n⊢ ↑n ≤ ENat.card V - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 78
} | {
"line": 56,
"column": 79
} | [
{
"pp": "case refine_4\nV : Type u_1\nG G' : SimpleGraph V\nx b a c : V\nM : (G ⊔ edge a c).Subgraph\np : G'.Walk a x\nhp : p.IsPath\nhcalt : G'.IsAlternating M.spanningCoe\nhM2nadj : ¬M.Adj x a\nhpac : p.toSubgraph.Adj a c\nhnpxb : ¬p.toSubgraph.Adj x b\nhM2ac : M.Adj a c\nhgadj : G.Adj x a\nhnxc : x ≠ c\nhnab... | [
"case refine_4\nV : Type u_1\nG G' : SimpleGraph V\nx b a c : V\nM : (G ⊔ edge a c).Subgraph\np : G'.Walk a x\nhp : p.IsPath\nhcalt : G'.IsAlternating M.spanningCoe\nhM2nadj : ¬M.Adj x a\nhpac : p.toSubgraph.Adj a c\nhnpxb : ¬p.toSubgraph.Adj x b\nhM2ac : M.Adj a c\nhgadj : G.Adj x a\nhnxc : x ≠ c\nhnab : a ≠ b\nhl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 132,
"column": 29
} | {
"line": 133,
"column": 42
} | {
"line": 133,
"column": 43
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nhveven : Even (Nat.card V)\nh : ¬G.IsTutteViolator G.universalVerts\nh' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp\nval✝ : Fintype V\nM : G.Subgraph\nhM : M.IsMatching\nhsub : M.vertsᶜ ⊆ G.univer... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nhveven : Even (Nat.card V)\nh : ¬G.IsTutteViolator G.universalVerts\nh' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp\nval✝ : Fintype V\nM : G.Subgraph\nhM : M.IsMatching\nhsub : M.vertsᶜ ⊆ G.universalVerts\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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